Theorem nnssn0s 28344

Description: The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
nnssn0s	⊢ ℕs ⊆ ℕ0s

Proof of Theorem nnssn0s

Step	Hyp	Ref	Expression
1		df-nns 28339	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
2		difss 4159	. 2 ⊢ (ℕ0s ∖ { 0s }) ⊆ ℕ0s
3	1, 2	eqsstri 4043	1 ⊢ ℕs ⊆ ℕ0s

Syntax hints:   ∖ cdif 3973   ⊆ wss 3976  {csn 4648   0_s c0s 27885  ℕ_0scnn0s 28336  ℕ_scnns 28337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-dif 3979  df-ss 3993  df-nns 28339

This theorem is referenced by:  nnssno  28345  nnn0s  28350  nnn0sd  28351  nnsgt0  28360